What is the vertex of the graph of $f(x)=|x+5|-6$?
A. $(-6,-5)$
B. $(-6,5)$
C. $(-5,-6)$
D. $(5,-6)$

The function is f ( x ) = ∣ x + 5∣ − 6 .
The x-coordinate of the vertex is found by setting the expression inside the absolute value to zero: x + 5 = 0 , so x = − 5 .
The y-coordinate of the vertex is found by substituting x = − 5 into the function: f ( − 5 ) = ∣ − 5 + 5∣ − 6 = − 6 .
The vertex of the graph is ( − 5 , − 6 ) ​ .

Explanation

Analyze the problem We are given the function f ( x ) = ∣ x + 5∣ − 6 and asked to find the vertex of its graph. The vertex of an absolute value function f ( x ) = ∣ x − h ∣ + k is the point ( h , k ) . In our case, we can rewrite the function as f ( x ) = ∣ x − ( − 5 ) ∣ + ( − 6 ) .

Find x-coordinate of the vertex The absolute value function reaches its minimum value when the expression inside the absolute value is equal to zero. So, we need to find the value of x for which x + 5 = 0 . Solving for x , we get x = − 5 .

Find y-coordinate of the vertex Now, we substitute x = − 5 into the function to find the y -coordinate of the vertex: f ( − 5 ) = ∣ − 5 + 5∣ − 6 = ∣0∣ − 6 = 0 − 6 = − 6 .

State the vertex Therefore, the vertex of the graph of f ( x ) = ∣ x + 5∣ − 6 is ( − 5 , − 6 ) .


Examples
Absolute value functions are used in various real-world scenarios, such as determining the distance from a target in engineering or physics. For example, if an engineer wants to ensure a part is within a certain tolerance, they might use an absolute value function to model the difference between the actual measurement and the desired measurement. Similarly, in economics, absolute value functions can model deviations from an expected price or value. Understanding the vertex helps to identify the point of minimum deviation or optimal performance.

Answered by GinnyAnswer | 2025-07-06